As we have seen in Chapter 1, the complete group ring Zp[[Γ]] ∼= Λ plays a fundamental role in the study of the arithmetic properties of Zp-extensions K∞/K. We want to generalise the construction given in Section 1.2 in order to be able to apply it to multiple Zp-extensions. Therefore we give the following very general definition.
Definition 2.9. Let G be a profinite group, i.e., a compact Hausdorff topo-logical group such that there exists a system of neighbourhoods of the neutral element containing only normal subgroups. Let O be a local ring with unique maximal idealpthat is Hausdorff and complete with respect to thep-adic topol-ogy. We furthermore assume thatOis compact. Then we define thecompleted group ring of G over O to be the topological inverse limit
O[[G]] := lim←−
U
O[G/U]
of the group ringsO[G/U], whereU runs through all the open normal subgroups of G.
2.2. GROUP RINGS AND POWER SERIES 35 Remarks 2.10.
(1) Let U ⊆ G be an open subgroup. Then we can write G as the union of pairwise disjoint cosets moduloU, i.e. G=S
iσi·U, whereσiruns through a system of representatives ofG/U. Since Gis compact and all the σi·U are open, we can conclude thatU is of finite index in G. Therefore every O[G/U] is the group ring of a finite group overO.
(2) For any profinite topological group G, we have an isomorphism (alge-braically and topologically)G∼= lim←−G/U, whereU runs through the open normal subgroups ofG (see [Neu 92], Theorem IV.2.8).
Here the projective limit is taken according to the canonical projection mappings induced by inclusions (i.e., the open normal subgroups of Gare ordered partially by inclusion; ifUi⊇Uj, then we consider the maps
fi,j :G/Uj −→ G/Ui
between finite groups). This projective system also induces the inverse limit lim←−O[G/U].
(3) The open normal subgroups ofZp are exactly the groupspnZpwithn∈N0
({0} is not open since Zp/(0) has infinite order). Therefore Definition 2.9 is a direct generalisation of the definition ofO[[Γ]] given in Section 1.2.
In the following, we will prove a generalisation of Theorem 1.9 for multiple Zp-extensions. We therefore will have to deal with rings of formal power series in several variables and coefficients inO. Before stating the theorem, we will collect some properties of such rings. This makes use of the following concepts.
In what follows, letOdenote an arbitrary ring. For any prime idealp⊆ O, we can consider the localisationOp. IfO is a domain, then eachOpis a subring of the quotient field ofO.
Definition 2.11. The height of p is defined to be ht(p) := dim(Op), where dim means theKrull dimension of the ringOp, i.e., the maximal lengthnof a chain of prime ideals
p0 $ p1 $ . . . $ pn
inOp. This corresponds to the maximal length of a chain of prime ideals in O descending fromp.
Be aware of the numbering which takes care of the trivial ideal {0}, which is prime ifOp is a domain.
Let P(O) denote the set of prime ideals p⊆ O of height 1.
Definition 2.12. Let nowO be a local ring with maximal ideal m.
(1) LetI ⊆ O be an ideal. Then we callI anideal of definition ofO if there exists an integerν >0 such thatmν ⊆I ⊆m.
(2) Letdbe the Krull dimension of O as defined in Definition 2.11. If I is an ideal of definition ofO that is generated bydelements x1, . . . , xd, then we say that{x1, . . . , xd} is asystem of parameters ofO.
(3) If there is a system of parameters that generates the maximal idealm, then we say that Ois a regular local ring.
(4) An arbitrary (not necessarily local) Noetherian ringO is called regular if for every prime ideal p⊆ O, the localisationOp is a regular local ring.
Definition 2.13. LetO be a domain with quotient fieldK. ThenO is called completely integrally closed if the following condition holds:
If x ∈ K is such that there exists a finitely generated O-submodule of K containing every power xn,n∈N, thenx∈ O.
Proposition 2.14. Let O be a domain.
(i) If O is completely integrally closed, thenO is integrally closed.
(ii) If O is Noetherian, then the converse of (i) holds.
(iii) If O is completely integrally closed, then O[X]and O[[X]] are completely integrally closed.
Proof. (i) and (ii): See [Bou 89], Chapter V,§1, no. 1 and no. 4..
(iii): See [Bou 89], Chapter V,§1, no.4, Proposition 14.
Lemma 2.15. LetOdenote a regular factorial local ring with maximal idealm.
Suppose that O is Hausdorff and complete with respect to the m-adic topology, and that the residue field O/m is finite. Let d∈N. Then the rings of formal power series in dvariables over O have the following properties:
(i) O[[T1, . . . , Td]] is a local ring with maximal ideal Md = m+ (T1, . . . , Td).
It is Hausdorff and complete with respect to the Md-adic topology.
(ii) O[[T1, . . . , Td]] is a compact topological group.
(iii) O[[T1, . . . , Td]] is a unique factorisation domain.
(iv) If O is Noetherian, then also O[[T1, . . . , Td]] is Noetherian.
(v) If O is Noetherian and integrally closed, then also O[[T1, . . . , Td]] is inte-grally closed.
(vi) If O is Noetherian and integrally closed, then we have O[[T1, . . . , Td]] = \
p∈P(O[[T1,...,Td]])
((O[[T1, . . . , Td]])p).
Proof. (i) It is a general fact that for a local ring A, the ringA[[T1, . . . , Td]]
of formal power series in a finite number of variables is local, too (see [Bou 89], Chapter II,§3, no. 1). Furthermore, using the corollary of Prop-osition 6 in [Bou 89], Chapter III, §2, no. 6, we inductively obtain that the maximal idealMdofO[[T1, . . . , Td]] is generated bymandT1, . . . , Td, and that O[[T1, . . . , Td]] is Hausdorff and complete with respect to the Md-adic topology.
(ii) Since O[[T1, . . . , Td]] is Hausdorff and complete with respect to the Md -adic topology, O[[T1, . . . , Td]] may be canonically identified with the in-verse limit of the finite discrete quotients (O[[T1, . . . , Td]])/Mid, i ∈ N, see [Bou 89], Chapter III, §2, no. 6. This limit is compact (see [Neu 92], Theorem IV.2.3).
2.2. GROUP RINGS AND POWER SERIES 37 (iii) Since O is a regular local ring, Theorem 19.5 of [Mat 86] implies that O[[T1, . . . , Td]] is regular, too. By a theorem ofAuslander and Buchs-baum (see Theorem 20.3 in [Mat 86]), every regular local ring is a unique factorisation domain.
(iv) IfA denotes any Noetherian domain, then also A[[T1, . . . , Td]] is Noethe-rian, see [Bou 89], Chapter III, §2, no. 10, Corollary 6.
(v) Since O is Noetherian and integrally closed, it is completely integrally closed by Proposition 2.14, (ii). The assertion follows inductively by using (iii) and, finally, (i) of the same proposition.
(vi) This is an immediate consequence of (iv) and (v), which together imply that O[[T1, . . . , Td]] is a so-called Krull domain, see [Bou 89], Corollary 1 to Lemma 1 in Chapter VII, §1, no. 3. The statement then follows from Theorem 4 in [Bou 89], Chapter VII, §1, no. 6.
We now specialise to the caseO=Zp(this will be enough for our purposes).
Definition 2.16. For any d ∈ N let Λd := Zp[[T1, . . . , Td]] denote the ring of formal power series in d variables having coefficients in Zp. In particular, Λ1 = Λ is the ring studied in Chapter 1.
Lemma 2.15 yields the following properties of the rings Λd. Proposition 2.17.
(i) Λdis a local ring with unique maximal ideal given byMd= (p, T1, . . . , Td).
It is Hausdorff and complete with respect to theMd-adic topology.
(ii) Λd is regular with Krull dimension equal to d+ 1.
(iii) Λd is a compact topological group.
(iv) Λd is a unique factorisation domain.
(v) Λd is Noetherian and integrally closed.
(vi) We have
Λd = \
p∈P(Λd)
((Λd)p).
Proof. Everything except (ii) follows immediately from Lemma 2.15. Since the ringZp is a Dedekind domain and therefore any prime idealp6= (0) is maximal, the Krull dimension ofZp is equal to 1. Since m= (p) is the maximal ideal of the local ringZp, we know that{p}is a system of parameters ofZp. Therefore Zp is a regular local ring (compare Definition 2.12, (3)). By Theorem 19.5 of [Mat 86], the ring of formal power series over a regular ring again is regular.
Using Theorem 15.4 in [Mat 86], we can compute the Krull dimension of Λd as follows:
dim(Zp[[T1, . . . , Td]]) = dimZp+d = d+ 1.
We now come to the generalisation of Theorem 1.9 announced above. Let K and K be as in Section 2.1, and write G = Gal(K/K) =< σ1, . . . , σd>Zp with fixed topological generatorsσ1, . . . , σd.
Theorem 2.18. Zp[[G]] ∼= Λd, the isomorphism of Zp-algebras (and homeo-morphism of topological groups) being induced by σi7→1 +Ti, i= 1, . . . , d.
Proof. The case d = 1 is covered by Theorem 1.9. We now let d ∈ N be arbitrary.
For every integern∈N0, we consider the subgroupGpn ⊆G generated by the elementsσp1n, . . . , σpdn, and we let Gndenote the quotient group
G/Gpn ∼= (Z/pnZ)d,
respectively. Then it is easy to see thatZp[[G]] is algebraically and topologically isomorphic to the projective limit lim←−Zp[Gn], where the limit is taken with respect to the projections πn,m :Gn −→ Gm,n ≥m, that are induced by the inclusions Gpn ⊆Gpm, respectively:
Indeed, by [Neu 92], Theorem IV.2.8,Gis isomorphic to the projective limit lim←−G/U, where U runs over the open normal subgroups of G; since G ∼= Zdp, these are isomorphic to
d
Q
j=1
pnjZp,nj ∈N0 for everyj= 1, . . . , d, and therefore every such U contains some Gpn. But then we have lim←−G/U ∼= lim←−Gn and Zp[[G]]∼= lim←−Zp[Gn].
For every fixed integern, there exists an isomorphism Zp[Gn] −→∼ Zp[T1, . . . , Td]/In,
where the idealIn⊆Zp[T1, . . . , Td] is generated by the elements (T1+ 1)pn−1, . . ., (Td+ 1)pn−1. Here the isomorphism is induced by mapping each generator σi∈Gn=G/Gpn ∼= (Z/pnZ)d to the polynomial (Ti+ 1)pn−1, respectively.
We therefore have to show that
Zp[[T1, . . . , Td]] ∼= lim←−Zp[T1, . . . , Td]/ (T1+ 1)pn−1, . . . ,(Td+ 1)pn−1 . By Proposition 2.17, (iii), Λd = Zp[[T1, . . . , Td]] is a compact topological group. The canonical projections Λd −→ Λd/In, n ∈ N, define a continuous homomorphism ϕ : Λd −→ lim←−Λd/In. Let Md := (p, T1, . . . , Td) denote the maximal ideal of Λd. Since
\
n≥0
In ⊆ \
n≥0
Mnd = {0}, the map ϕis injective.
Let (fn)n≥0 ∈ lim←−Λd/In denote an arbitrary element; we will show that there exists a pre-imagef ∈Λdunderϕ: For eachn, we choose a representative fn ∈ Λd of fn ∈ Λd/In. Since Λd is complete with respect to the Md-adic topology (see Proposition 2.17, (i)), and sinceIn⊆Mnd for everyn, there exists an element f ∈ Λd such that f ∈ T
n≥0
fn = T
n≥0
fn ·In (note that for every j ≥i, we have fj ≡fi mod Ii). But then ϕ(f) = (fn)n, and thereforeϕ is an isomorphism.
2.2. GROUP RINGS AND POWER SERIES 39 Furthermore, every quotient
Λd/In ∼= Zp[T1, . . . , Td]/In ∼= Zd·pp n
is profinite, and therefore also the limit lim←−Λd/Inis a profinite group (compare Lemma 1.2.6, (c) in [FJ 08]), and in particular Hausdorff. Since Λdis compact, it follows thatϕ: Λd−→lim←−Λd/Inis a homeomorphism (see [Os 92], Corollary 2.4.9).
We will conclude this section by giving an overview of the theory of Λd -modules (analogously to the theory of Λ--modules described in Section 1.2, which culminated in the Structure Theorem 1.24 – see Theorem 2.23 below).
Definition 2.19. A finitely generated Λd-moduleM is calledpseudo-null if Mp ={0} for all prime ideals p⊆Λd of height≤1.
Remarks 2.20.
(1) A pseudo-null Λd-moduleM is Λd-torsion.
(2) M is pseudo-null if and only if it satisfies the following equivalent condition:
Ifp is a prime ideal with Ann(M)⊆p, then ht(p)≥2. Here Ann(M) ={x∈Λd|x·M ={0}}
denotes the annihilator ideal ofM.
(3) IfM is pseudo-null, then (2) implies thatM is annihilated by two relatively prime elements of Λd.
In fact, ifJ := Ann(M), and if 06=g∈J is arbitrary, then there exists an elementh∈J coprime to g:
Let 06=g∈J be arbitrary, and writeg=
r
Q
i=1
peii, with irreducible elements pi in the unique factorisation domain Λd (compare Proposition 2.17, (iv)).
For everyi= 1, . . . , r, there exists an elementhi∈J such thatpi -hi, since otherwise,J would be contained in the prime ideal (pi)⊆Λdof height one.
Without loss of generality, we may assume that pj | hi for every j 6= i.
Theng is coprime to h:=h1+. . .+hr ∈J.
(4) A Λ1= Λ-module is pseudo-null if and only if it is finite.
Proof. See the remarks after Definition 5.1.4 in [NSW 08]; for (4) we use that Λ1 = Λ is a 2-dimensional, Noetherian, integrally closed local domain with finite residue fieldZp[[T]]/(p, T) ∼= Z/pZ; compare Proposition 2.17, (i), (iv) and (v).
Definition 2.21. A homomorphism f : M −→ N of finitely generated Λd -modules is called apseudo-isomorphism if the kernel and cokernel of f are pseudo-null Λd-modules. Equivalently, this is the case if we have an exact sequence
0−→M1−→M −→f N −→M2−→0
with pseudo-null Λd-modules M1 and M2. We writeM ∼N if there is such a pseudo-isomorphism.
Remarks 2.22.
(1) In general, M ∼ N does not imply N ∼ M (compare Remarks 1.20, (1) for an example in the case d= 1). But if M and N are finitely generated torsion Λd-modules, then M ∼ N if and only if N ∼ M, see the remarks on page 271 of [NSW 08].
(2) In view of Remarks 2.20, (4), the notion of pseudo-isomorphic Λ1-modules introduced here coincides with the definition given in Chapter 1 (see Defi-nition 1.19).
Theorem 2.23 (Structure Theorem). Let M be a finitely generated Λd -module. Then there exist an integers∈N0, finitely many prime idealsp1, . . . ,ps
where FΛd(M) denotes the maximal torsion-free quotient of M. The prime ideals pi and the numbers ni are uniquely determined by M.
For d= 1, we can replace the module FΛd(M) by a free Λ-module, i.e., there exists an integer r∈N0 such that we have a pseudo-isomorphism
f :M −→ Λr ⊕
s
M
i= 1
Λd/pnii .
Proof. By [NSW 08], Proposition 5.1.7, we have a pseudo-isomorphism f :M −→ FΛd(M) ⊕
s
M
i= 1
Λd/pnii .
For d = 1, compare Theorem 1.24 or see [NSW 08], Propositions 5.1.8 and 5.1.9. gen-erated by irreducible elements gi ∈ Λd, respectively. Indeed, let 0 6=x be contained in a prime ideal p ⊆Λd of height one. We write x as a product of irreducible elements in the unique factorisation domain Λd. Since p is a prime ideal, at least one irreducible divisor gof xhas to be contained inp.
But then (g) ⊆ Λd is a prime ideal contained in p, and therefore (g) =p, because pis of height one.
(2) If E denotes an elementary Λd-module, thenE does not contain any non-trivial pseudo-null submodules.
2.3. GREENBERG’S TOPOLOGY 41